Skip to main content
SpringerLink
Log in

On the Black-Box Complexity of Sperner’s Lemma

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

We present several results on the complexity of various forms of Sperner’s Lemma in the black-box model of computing. We give a deterministic algorithm for Sperner problems over pseudo-manifolds of arbitrary dimension. The query complexity of our algorithm is linear in the separation number of the skeleton graph of the manifold and the size of its boundary. As a corollary we get an \(O(\sqrt{n})\) deterministic query algorithm for the black-box version of the problem 2D-SPERNER, a well studied member of Papadimitriou’s complexity class PPAD. This upper bound matches the \(\Omega(\sqrt{n})\) deterministic lower bound of Crescenzi and Silvestri. The tightness of this bound was not known before. In another result we prove for the same problem an \(\Omega(\sqrt[4]{n})\) lower bound for its probabilistic, and an \(\Omega(\sqrt[8]{n})\) lower bound for its quantum query complexity, showing that all these measures are polynomially related.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aaronson, S.: Lower bounds for local search by quantum arguments. SIAM J. Comput. 35(4), 804–824 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aaronson, S., Shi, Y.: Quantum lower bounds for the collision and the element distinctness problems. J. ACM 51(4), 595–605 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ambainis, A.: Polynomial degree vs. quantum query complexity. J. Comput. Syst. Sci. 72, 220–238 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. J. ACM 48(4), 778–797 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Beame, P., Cook, S., Edmonds, J., Impagliazzo, R., Pitassi, T.: The relative complexity of NP search problems. J. Comput. Syst. Sci. 57(1), 3–19 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bennett, C., Bernstein, E., Brassard, G., Vazirani, U.: Strength and weaknesses of quantum computing. SIAM J. Comput. 26(5), 1510–1523 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bloch, E.: Mod 2 degree and a generalized No Retraction Theorem. Math. Nachr. 279(5), 490–494 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Buresh-Oppenheim, J., Morioka, T.: Relativized NP search problems and propositional proof systems. In: 19th Conference on Computational Complexity, pp. 54–67 (2004)

  9. Chen, X., Deng, X.: On the complexity of 2D discrete fixed point problem. In: 33rd International Colloquium on Automata, Languages and Programming, pp. 489–500 (2006)

  10. Chen, X., Teng, S.: Paths beyond local search: a nearly tight bound for randomized fixed-point computation. In: 48th Annual Symposium on Foundations of Computer Science, pp. 124–134 (2007)

  11. Chen, X., Sun, X., Teng, S.: Quantum separation of local search and fixed point computation. In: 14th Annual International Computing and Combinatorics Conference (2008, to appear)

  12. Crescenzi, P., Silvestri, R.: Sperner’s lemma and robust machines. Conf. Comput. Complex. 7(2), 163–173 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. R. Soc. A 439, 553–558 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fan, K.: Simplicial maps from an orientable n-pseudomanifold into S m with the octahedral triangulation. J. Comb. Theory 2, 588–602 (1967)

    Article  MATH  Google Scholar 

  15. Friedl, K., Ivanyos, G., Santha, M., Verhoeven, Y.: On the black-box complexity of Sperner’s Lemma. In: 15th International Symposium on Fundamentals of Computation Theory. LNCS vol. 3623, pp. 245–257 (2005)

  16. Friedl, K., Ivanyos, G., Santha, M., Verhoeven, Y.: Locally 2-dimensional Sperner problem complete for the Polynomial Parity Argument classes. In: 6th Italian Conference on Algorithms and Complexity. LNCS vol. 3998, pp. 380–391 (2006)

  17. Gilbert, J., Hutchinson, J., Tarjan, R.: A separator theorem for graphs of bounded genus. J. Algorithms 5(3), 391–407 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  18. Grigni, M.: A Sperner Lemma complete for PPA. Inf. Process. Lett. 77(5–6), 255–259 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Laplante, S., Magniez, F.: Lower bounds for randomized and quantum query complexity using Kolmogorov arguments. SIAM J. Comput. 38(1), 46–62 (2008). 19th Conference on Computational Complexity, pp. 294–304 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Llewellyn, D., Tovey, C.: Dividing and conquering the square. Discrete Appl. Math. 43(2), 131–153 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  21. Llewellyn, D., Tovey, C., Trick, M.: Local optimization on graphs. Discrete Appl. Math. 23(2), 157–178 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Megiddo, N., Papadimitriou, C.: On total functions, existence theorems and computational complexity. Theor. Comput. Sci. 81, 317–324 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  23. Papadimitriou, C.: On graph-theoretic lemmata and complexity classes. In: 31st Foundations of Computer Science, pp. 794–801 (1990)

  24. Papadimitriou, C.: On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. System Sci. 48(3), 498–532 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  25. Santha, M., Szegedy, M.: Quantum and classical query complexities of local search are polynomially related. In: 36th Symposium on Theory of Computing, pp. 494–501 (2004). To appear in Algorithmica

  26. Simon, D.: On the power of quantum computation. SIAM J. Comput. 26(5), 1474–1783 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sperner, E.: Neuer Beweis für die Invarianz der Dimensionzahl und des Gebietes. Abh. Math. Semin. Hamb. Univ. 6, 265–272 (1928)

    Article  MATH  Google Scholar 

  28. Špalek, R., Szegedy, M.: All quantum adversary methods are equivalent. Theory Comput. 2(1), 1–18 (2006)

    MathSciNet  Google Scholar 

  29. Taylor, L.: Sperner’s Lemma, Brouwer’s Fixed Point Theorem, The Fundamental Theorem of Algebra. http://www.cs.csubak.edu/~larry/math/sperner.pdf

  30. Thomason, A.: Hamilton cycles and uniquely edge colourable graphs. Ann. Discrete Math. 3, 259–268 (1978)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Budapest University of Technology and Economics, P.O. Box 91, 1521, Budapest, Hungary

    Katalin Friedl

  2. Computer and Automation Research Institute, Hungarian Academy of Sciences, P.O. Box 63, 1518, Budapest, Hungary

    Gábor Ivanyos

  3. CNRS–LRI, Université Paris-Sud, 91405, Orsay, France

    Miklos Santha

  4. Centre for Quantum Technologies, National University of Singapore, Singapore, 117543, Singapore

    Miklos Santha

  5. LRI, Université Paris-Sud, 91405, Orsay, France

    Yves F. Verhoeven

  6. ENST, 46 rue Barrault, 75013, Paris, France

    Yves F. Verhoeven

Authors
  1. Katalin Friedl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gábor Ivanyos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Miklos Santha
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yves F. Verhoeven
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miklos Santha.

Additional information

Research supported by the European Commission IST Integrated Project Qubit Application (QAP) 015848, the OTKA grants T42559 and T46234, and by the ANR Blanc AlgoQP grant of the French Research Ministry.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Friedl, K., Ivanyos, G., Santha, M. et al. On the Black-Box Complexity of Sperner’s Lemma. Theory Comput Syst 45, 629–646 (2009). https://doi.org/10.1007/s00224-008-9121-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-008-9121-2

Keywords

Search

Navigation